Problem: The local linear approximation to the function $h$ at $x=0$ is $y=4x-5$. What is the value of $h(0)+h'(0)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $-3$ (Choice B) B $-2$ (Choice C) C $-1$ (Choice D) D $0$
Solution: The local linear approximation of $h$ at $x=0$ is achieved using the equation of the line tangent to $h$ at $x=0$. In other words, $y=4x-5$ is the equation of the line tangent to the graph of $h$ at $x=0$. How can we use this to find $h(0)$ and $h'(0)$ ? Since the line is tangent to the graph of $h$ at $x=0$, we know two key facts about it: The line passes through the point $({0},{h(0)})$ The line's slope is ${h'(0)}$ The slope of $y={4}x-5$ is ${4}$. The $y$ -value that corresponds to $x={0}$ is $4({0})-5={-5}$. Now we can find our answer: ${h(0)}+{h'(0)}={-5}+{4}=-1$